A pure tone of around 300 Hz is found in the pipework of a ventilation system, and it needs to be controlled. Design a side-branch Helmholtz resonator that can reflect the noise back to its source. The duct has a square cross-section of 20 cm by 20 cm, and the neck of the resonator is formed by a circular hole of r=1cm in radius through the duct wall of 2 mm in thickness. The virtual length associated with the air motion through this hole is approximately 1.7r. Damping is ignored.

(a) If two more tones are detected at 200 Hz and 400 Hz, explain why expansion chamber is a better choice for noise reflection.

(b) Then proceed to design one such device by specifying the chamber length and the area expansion ratio if all three tones are to be reduced by 10 dB.

Ans: L=0.2833m, a=7.07